Combinatorics and Geometry in Representation Theory

该奖项支持组合学和几何学的研究，并将其应用于表示理论。该项目研究反射群的几何和不变理论。PI揭示了复多面体、类Coxeter复形、Hecke代数和余不变代数之间的关系。这些技术可能有助于发展复反射群的Kazhdan-Lusztig胞元理论。例如，立方体或十二面体(或任何柏拉图实体)的对称构成一个反射群。这些群是由镜像反射(关于超平面)产生的，出现在物理、生物、化学和计算机科学中。反射群也有着丰富的历史，连接了许多不同的数学领域：离散几何、拓扑学、奇点理论、超平面的排列、Lie理论、组合学和不变量理论。

This award supports research in combinatorics and geometry withapplications to representation theory. The project investigates thegeometry and invariant theory of reflection groups. The PI exploresrelations between complex polytopes, Coxeter-like complexes, Heckealgebras, and the coinvariant algebra. Techniques may help develop aKazhdan-Lusztig theory of cells for complex reflection groups.Reflection groups arise in nature as symmetry groups. For example, thesymmetries of the cube or dodecahedron (or any Platonic solid) forma reflection group. These groups are generated by mirror reflections(about hyperplanes) and appear in physics, biology, chemistry, andcomputer science. Reflection groups also have a rich history ofconnecting many different areas of mathematics: discrete geometry,topology, singularity theory, arrangements of hyperplanes, Lietheory, combinatorics, and invariant theory.